Rates of Convergence for Distributed Average Consensus with Probabilistic Quantization

Probabilistically quantized distributed averaging (PQDA) is a fully decentralized algorithm for performing average consensus in a network with finite-rate links. At each iteration, nodes exchange quantized messages with their immediate neighbors. Then each node locally computes a weighted average of the messages it received, quantizes this new value using a randomized quantization scheme, and then the whole process is repeated in the next iteration. In our previous work we introduced PQDA and demonstrated that the algorithm almost surely converges to a consensus (i.e., every node converges to the same value). The present article builds upon this work by characterizing the rate of convergence to a consensus. We illustrate that the rate of PQDA is essentially the same as unquantized distributed averaging when the discrepancy among node values is large. When the network has nearly converged and all nodes’ values are at one of two neighboring quantization points, then the rate of convergence slows down. We bound the rate of convergence during this final phase by applying lumpability to compress the state space, and then using stochastic comparison methods.